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  Understanding Tests and Measurements for the Parent and Advocate

Peter W. D. Wright, Esq. & Pamela Darr Wright, M.A., M.S.W. Licensed Clinical Social Worker 2001

"If something exists, it exists in some amount.
If it exists in some amount, then it is capable of being measured."
Rene Descartes, Principles of Philosophy, 1644

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I. Introduction
A. Background
B. Evidence & Law
C. Objective Measurement of Progress

II. The Process of Educational Decision-Making
A. Katie
B. Measurement of Change: Rulers, Yardsticks and Other Tools
C. What Do Evaluations Tell You?

III. Statistics: General Principles
A. Statistics: Applied

IV. Measuring Progress: The Bell Curve
A. Understanding The Bell Curve
B. Composite Scores
C. Subtest Scatter
D. When Apparent Progress Means Actual Regression
C. Norm Referenced versus Criterion Referenced Tests
D. Standard Deviation
E. Standard Scores
F. Rules to Remember

V. Uunderstanding Test Data
A. Other Tests: Means & Standard Deviations
B. Specific Tests
C. Other Tests
D. Private Evaluations
E. Individualized Education Programs (IEPs)

VI. Parents "To-Do" List

Learn More About Tests and Assessments, View our New Slide Show : Educational Progress Graphs 

About the Author  


Most parents of special needs children know that they must understand the law and their rights. Few parents know that they must also understand the facts. The "facts" of their child's case are contained in the various tests and evaluations that have been administered to the child. Changes in test scores over time provide the means to assess educational benefit or regression. Most important educational decisions, from eligibility to the intensity of educational services provided, are based on the results of psychological and educational achievement testing. Parents who obtain appropriate special educational programs for their children have learned what different tests measure and what the test results mean. 

As an attorney who specializes in representing special education children, I know that many parents consult with me after deciding that their child's special education program is not appropriate. These parents are often right. However, in most cases they do not have the evidence to support their belief, nor do they know how to interpret and use the evidence contained in educational and psychological tests. They need evidence to support their beliefs. 

Often these parents are convinced that a special education program is not providing sufficient help for the child --- that under the present special education program, the child is failing to make progress and has fallen further behind. These parents experience a sense of urgency --- the child has usually received special education for several years and time is running out. 

Critical educational decisions are often made, based on the subjective beliefs of parents and educators. As a parent, you may believe that your child is not making adequate progress in a special education program. The special education staff may firmly believe that he is doing as well as he can --- or that your expectations are too high. Without objective information, both sides will take positions that are based upon emotions --- and tempered by hopes and fears. Effective educational decision-making must be based on objective information and facts, not subjective emotional reactions and beliefs. 

Before you can participate in the development of an appropriate special education program, you must have a thorough understanding of your child's strengths and weaknesses. This information is contained in the various tests that are used to measure the child's ability and educational achievement. 

To successfully advocate for your child, you must also learn about tests and measurements --- statistics. Statistics are ways of measuring progress or lack of progress, using numbers. After you analyze the scores your child obtains when tested and understand what these numbers mean, you will be able to develop an appropriate educational program for your child --- a program from which the child benefits. 

As you master the material contained in this article, you will understand what various tests and evaluations measure and how to use information from tests to measure academic progress. You will learn how to use graphs to visually demonstrate your child's progress or lack of educational progress in a very powerful and compelling manner.

The United States Supreme Court
Florence County School District Four v. Shannon Carter
November 9, 1993

In Florence County School District Four v. Shannon Carter, 510 U. S.7, 114 S. Ct. 361, (1993), the United States Supreme Court issued a landmark decision. In Carter, the school system defaulted on their obligation to provide a free appropriate education to Shannon Carter, a child with learning disabilities and an Attention Deficit Disorder. Let's look at how the courts viewed the facts and the law in the Carter case.


When Shannon was in the seventh grade, her parents talked to the public school staff and expressed concerns about Shannon's reading and academic problems. She was evaluated by a public school psychologist who described Shannon as a "slow learner" who was lazy, unmotivated and needed to be pressured to try harder. Her parents pressured her to work harder. Despite the intense pressure, when Shannon was in the ninth grade, she failed several subjects. Her parents had her evaluated by a child psychologist. That evaluator determined that Shannon's intellectual ability was actually above average. Educational achievement testing demonstrated that sixteen year old Shannon was reading at the fifth grade level (5.4 GE) and doing math at the sixth grade level (6.4 G.E.). Shannon had dyslexia. As she prepared to enter tenth grade, she was also functionally illiterate. 

In Shannon's case, the school district developed an IEP which proposed that after a year of special education in the tenth grade, Shannon would read at the 5.8 grade equivalent level and perform math at the 6.8 grade equivalent level. In other words, after one year of special education designed to remediate her learning disabilities, Shannon was expected to gain only four tenths of a year, as measured by her scores on the Woodcock-Johnson and KeyMath educational achievement tests, a gain from 5.4 to 5.8 and 6.4 to 6.8 grade levels in reading and math respectively. 

Shannon's parents insisted that their daughter required a more intensive program so that she could master necessary reading, writing and math skills. They felt that the proposed program was inadequate, and worried that Shannon would still be functionally illiterate when she graduated in three years. Emory Carter insisted that his daughter should be able to read, write and do arithmetic at a twelfth grade level when she graduated from high school. 

Although Emory and Elaine Carter shared their concerns and wishes with the public school officials, the administrators took a "take it or leave it" position and refused to provide Shannon with a more intensive special education program that provided actual remediation in reading, writing, and arithmetic. Seeking more services for their daughter, the parents requested a special education due process hearing. The Hearing Officer ruled that the public school IEP was appropriate. The parents appealed this decision to a Review Panel and lost again. 

At that point, Emory and Elaine Carter withdrew Shannon from her local public high school and enrolled her in Trident Academy. Trident is a private school in Mt. Pleasant, South Carolina that specializes in remediating children with learning disabilities, including dyslexia. Shannon's parents then appealed the Review decision to the U. S. District Court. They asked Judge Houck to award them reimbursement for Shannon's private school education at Trident. 

When Shannon graduated from Trident Academy three years later, her reading and math scores were on a high school level.

After hearing testimony and reviewing the transcripts and documents from the Due Process and Review Hearings, U. S. District Court Judge Houck found that the school district's IEP was "wholly inadequate" to meet Shannon's needs. He ruled that Shannon had received an appropriate education at Trident and ordered Florence County to reimburse Shannon's parents for the costs of her education. 

On what basis did Judge Houck decide that the IEP proposed by Florence County was inappropriate? What evidence caused him to decide that Shannon received an appropriate education at Trident Academy?

Evidence & Law

The decisions in Shannon's case, and in many special education cases, rest on the evidence provided by various tests and evaluations of the individual child. When Judge Houck wrote that the Florence County's IEP was "wholly inadequate" to meet Shannon's needs, he was relying on his interpretation of the results of testing. Judge Houck understood the importance of accurately interpreting test scores. He charted Shannon's test scores and included this data as part of his U. S. District Court decision. (See also Hall v. Vance, 555 EHLR 437, (E.D. NC 1983), affirmed at 774 F. 2d 629, 557 EHLR 155, (4th Cir. 1985)) in which U. S. District Court Judge Dupree charted out James Hall's test scores to support his 1983 decision that Vance County, North Carolina did not provide James with an appropriate education in the public school program.) When you finish this article, you will also be able to interpret and chart your child's test scores and measure educational progress or lack of progress. 

Florence County appealed Judge Houck's decision to the U. S. Circuit Court of Appeals for the Fourth Circuit. Appeals from the U. S. District Courts in Maryland, Virginia, West Virginia, North Carolina and South Carolina are heard in the U. S. Court of Appeals for the Fourth Circuit by a three judge panel. The Fourth Circuit is composed of thirteen judges. Appeals from U. S. Circuit Courts of Appeals are filed in the U. S. Supreme Court. Occasionally a U. S. Circuit Court of Appeals will convene all Judges appointed to the Circuit to hear a case. This is called an en banc review. 

A three judge panel of the Fourth Circuit affirmed Judge Houck's decision as to the inadequacy of Florence County's proposed IEP. Florence County then appealed to the United States Supreme Court. 

On November 9, 1993, the United States Supreme Court issued a unanimous decision on Shannon's behalf. In the Carter decision, authored by Justice Sandra Day O'Connor, the Court upheld the lower decisions, ruled against Florence County School District Four, and ordered them to reimburse Shannon's parents for the costs of her tuition, room and board, and attorney's fees.

Objective Measurement of Progress

IEPs must include objective means of measuring the child's progress in a special education program. Volume 34 of the Code of Federal Regulations, Section 300.347, "Content of individualized education program," states that an IEP must include:

(2) A statement of measurable annual goals, including benchmarks or short-term objectives, related to (i) Meeting the child’s needs that result from the child’s disability to enable the child to be involved in and progress in the general curriculum . . .


(7) A statement of (i) How the child’s progress toward the annual goals described in paragraph (a)(2) of this section will be measured; . . . 

In Shannon's case, her IEP stated that she "will be able to improve total reading level from the 5.4 grade level to the 5.8 grade level as measured by the Woodcock Reading Mastery Test . . . (and that she) will improve math skills from the 6.4 grade equivalent to the 6.8 grade equivalent as measured by the Key Math Diagnostic Test." This IEP complied with regulation in existence at that time, (34 C.F.R. § 300.346, since modified), by including "appropriate objective criteria." The criteria required a re-administration of the Woodcock-Johnson and KeyMath tests to measure progress. 

The U. S. District Court and the Fourth Circuit found that the proposed gain of four months after a year of special education was "wholly inadequate." 

In an effort to avoid Florence County's fate, many school districts around the country now develop IEPs that include no objective measures of the child's progress. Instead of including educational goals where the child's progress is measured using objective tests and measurements, as Florence County did with Shannon, many schools now propose IEPs that rely exclusively on subjective teacher observations of the child's progress. Let's see how this works. 

We'll look at Johnny, a child who has a learning disability that is manifested in the area of reading. Johnny is below grade level in reading. Instead of developing an IEP that will measure progress in reading on a specific objective test, the special education staff may come up with a goal such as: "Johnny will make measurable progress in reading, as measured by teacher observation and teacher made tests at 80% accuracy." 

"Objective measurement of progress" becomes the teacher's subjective observation as to whether the child has improved in reading, writing, or arithmetic. The criteria of mastery becomes 80% of a subjective opinion. When parents object and ask for a more intense program with clear independent objective standards, they are often rebuffed or criticized. 

Many school board counsel and state departments of education have advised schools to move away from using objective measurements of progress for special education children. 

If you believe that the special education your child is receiving is inadequate, you must have evidence to support your position. You will find this evidence in the public school and private sector testing that has been or will be completed on your child. 

After you master the material contained in this article, you will understand what the various tests and evaluations measure and how the test results are reported. You will know how to convert the scores on different tests into numbers that are easily understood. And, you will know how to measure educational progress or lack of progress, i.e. regression.


Three years ago, your eight year old son Mike began to have serious difficulties in school. By the time he reached third grade, his difficulty in reading was of great concern. His handwriting was nearly illegible and homework was a nightmare. On several occasions, you consulted with Mike's teacher about the problems he was having. Eventually, the teacher sent Mike's "case" to a special education committee. You attended a meeting of this committee --- which recommended that Mike be evaluated through the school's special education department. Relieved that something was going to be done, you consented to these battery of tests. 

According to the evaluations, your son has a learning disability. In Mike's case, he has visual-perceptual problems and visual-motor problems that negatively affect his ability to read and write. Based on the results of the evaluations, your son was found eligible for special education services through his neighborhood school. 

After Mike was found eligible for special education, you attended a meeting to develop his Individualized Education Program (IEP). This IEP provided for Mike to receive one period of special education in an "LD Resource" class every day. It was your understanding that Mike would receive individualized help in reading and writing from a teacher who was specially trained to remediate his learning disability problems. 

Three years have passed. Mike hasn't made much progress, despite the special education help. He still has difficulty reading aloud. His spelling is poor, and his handwriting is unreadable. He is behind most of the children in his class. His attitude has changed. He is angry and depressed and says he "hates school." 

When you discussed your concerns about Mike's lack of progress with his special education teacher, she reassured you that he was making progress and told you to be patient. You think that patience is not the issue; you are worried that your son will never master basic educational skills. What kind of future will he have? 

At a recent IEP meeting, you reiterated your concerns about Mike's lack of progress and expressed the belief that he needs more help than he is getting in the Resource program. The committee disagreed with you. One person told you that Mike was getting all the help he needs and that he was really doing quite well. Another committee member told you that your expectations were too high --- and that if you didn't accept Mike's limitations, you would damage him emotionally. 

What should you do? You know that the time in the LD resource class with several other children is not providing Mike with the individualized help he needs. The school has not focused on teaching your son how to read, write and do arithmetic. Now, the IEP team suggests more "accommodations" and "modifications." They propose to reduce his workload, give him untimed tests, and provide him with "talking books" and a calculator. They do not propose to give him individualized help so that he will learn to read, write, and do arithmetic. 

You believe that Mike's emerging "emotional problems" are due to shame and embarrassment about not being successful in school. How can you, a parent, prove this to the staff at Mike's school so that they will develop an appropriate educational program for him? How will you know when he is getting the help he needs?

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The Process of Educational Decision-Making 

Many parents erroneously assume that interpreting test data is beyond their competence and is the responsibility of the school personnel. If parents default on their responsibility and obligation to understand this information, then the interpretation of the test data is left to the school psychologist --- a person who often has very limited information about your child, aside from test scores.

The basic principles of tests and measurements are not difficult to master. As you read this article, you will see that you are already familiar with many of the concepts discussed. Statistics and statistical terms are used in many other areas of life, from business and sports to medicine. Newspaper and magazine articles use statistics to inform readers of change or lack of change. You read articles about changes in the population, the economy --- even public opinion polls --- that include statistical information to inform you or persuade you of a point.

Parents need to expend time and effort to develop an adequate degree of expertise in statistics. You should reread parts of this article several times. Underline, make margin notes, and use a highlighter to help you master the material. Be patient and put in the time. The time you expend will help to change your child's life.

As you study this material, you will probably encounter some terms and concepts that seem confusing at first --- terms like standard deviation, standard score, and grade and age equivalents. Other concepts will be familiar --- averages, percentiles.

After you master this information, you will understand the educational and psychological tests that are administered to your child. You will be able to use this information to make wise educational decisions. You will find that your newfound knowledge and expertise exceeds that of many of the special education committee members.

When you attend your next IEP or Eligibility meeting, you will be glad you did your homework!


Katie is a fourteen year old ninth grader. She "hates school" and is failing several subjects. As a young child, Katie was bright, happy, and curious. When she entered third grade, her attitude began to change. Now, she locks herself in her room, lies on her bed, and listens to music for hours. She is sullen and angry and says she can't wait to quit school.

In desperation, Katie's parents took her to a child psychologist for testing. At a meeting to interpret the test results to Katie and her parents, the psychologist explained that Katie scored two "standard deviations" above the mean on the Similarities subtest of the Wechsler Intelligence Test for Children, Third Edition (WISC-III) and two and a half "standard deviations" below the mean on the spontaneous writing sample of the Test of Written Language, Third Edition (TOWL-III).

Test publishers are constantly updating and revising their tests. The Wechsler Intelligence test for children was originally known as the WISC. Later, it was revised and became known as the WISC-R. Several years ago, the next version was published as the WISC-III. The first Test of Written Language (TOWL) was replaced by the TOWL-II and was recently revised again.

The Woodcock Johnson battery of tests was known as the Woodcock Johnson Psycho-Educational Battery. The WJPEB included educational achievement testing and cognitive ability testing. Dr. Woodcock also produced the Woodcock Reading Mastery Test. Today, the current test series is called the Woodcock-Johnson Psycho-Educational Battery, Revised, (WJ-R) which is an educational achievement test that includes the Test of Cognitive Abilities.

The current version of any popular test is probably in a revision status. A competitor test publishing company is probably trying to develop a new and better version of the competitor's product. This article will not focus on an analysis of each test's strengths and weaknesses. Weaknesses in a current test will probably be eliminated by the next version which will be out within a couple of years.

Parents must understand that tests do not necessarily measure what they purport to measure. As you will see, a child's score on a push-up test can be represented as an overall fitness score, a measure of arm strength, an upper body measurement score, a measure of perseveration and persistence, or a measure of a child's motivation. A score may measure only one of the variables or it may accurately reflect all of the above.

To demonstrate this point, let's look at tests that measure reading ability. One test that measures a child's reading ability actually measures the child's ability to correctly read aloud and pronounce isolated words out of context, i.e., a word recognition test. The test includes a list of words, i.e., cat, tree, dog, house, person, etc. This kind of reading test does not measure true reading and may be adversely impacted by speech or word finding problems.

Another reading test measures reading by having the child read a passage of text, then answer a series of multiple choice questions about the passage. In this case, the child's score may be a measure of the child's ability to intellectually eliminate certain answers of the multiple choice format, i.e., a test of reasoning, not true reading. Some very bright children may need to recognize and interpret only a few words to discern the total context. Other children have excellent word recognition abilities but cannot link or interpret the words in a body of text or passage. Another reading test has the child read a passage of text aloud (measuring oral reading) and then answer questions. The accuracy of the words read aloud and the child's understanding of the passage makes up the reading score.

You need to know exactly how the test was administered and what it measured.

When we first discussed Katie, we saw that she scored two "standard deviations" above the mean on the Similarities subtest of the Wechsler Intelligence Test for Children, Third Edition (WISC-III) and two and a half "standard deviations" below the mean on the spontaneous writing sample of the Test of Written Language, Third Edition (TOWL-III).

Do these test scores explain the academic problems Katie is having? Do they have anything to do with her moodiness and her intense dislike of school? (Answers: Yes and Yes.) When we return to Katie's case later in this article, you will understand the significance of her test scores. You will also understand why Katie's self esteem has plummeted.

Remember: After you master the material contained in this article, you will understand and be able to interpret your child's test scores. You will be able to go back to the preceding paragraph and understand the significance of Katie's scores. You will have acquired skills that will enable you to answer questions like these:

  • How is your child functioning, compared with other children the same age ?
  • How is your child functioning, compared with others in the same grade?
  • How much educational progress has your child made (what has been learned) since the last test battery?
  • If your child is receiving special education, has the child progressed or regressed in the special education placement?
  • If your child has shown an increase in age and grade equivalent test scores, has the child actually fallen further behind the peer group?
And, you will learn how to incorporate objective measurements into your child's IEP so that educational progress can be charted on a regular basis.

Measuring Change: Rulers, Yardsticks and Other Tools

To clarify these points, let's change the facts. You can measure your child's physical growth with a measuring tape and a bathroom scale. You can measure growth by charting how much height increases, as measured in inches, and how much weight increases, as measured by pounds, over a period of months or years. Using these tools, you can document his physical growth. You don't need to be a doctor to understand that increases in these measurements prove that your child is growing.

Assume that your child's height was five feet, three inches last year. This year, the child is five feet, six inches tall. You can report this information in several ways. You can say that last year, your child was sixty-three inches tall and is now sixty-six inches tall. Or, you can say that your child was 5.25 feet tall and is now five and a half feet tall. You can even say that a year ago, your child was 160 centimeters tall and is now 168 centimeters tall. Or, that your child was 1.75 yards tall and is now 1.83 yards tall!

If you (or your child's pediatrician) have been measuring your child at regular intervals, you can create a chart or graph that documents changes in height or weight over time. Your child's pediatrician has "growth charts" that you can use to compare your child's growth with the growth of the "average" child.

Likewise, educational growth can be measured and charted. The yardsticks used for measurement are different, but the principles are the same. Measuring educational growth or progress is not much different from charting physical growth. Instead of a tape measure and a set of bathroom scales, you need psychological and educational achievement test results. Where will you find the information you need? How can you measure change?

Most school districts test their students on standardized group educational achievement tests at regular intervals. The results of these tests provide information about how well school districts are accomplishing their mission of educating children. The information contained in the group standardized tests can provide you with some basic information.

Standardized educational achievement tests are general measures. The information they provide is similar to that provided by medical screening tests. Medical screening tests can suggest that a problem exists. Additional testing is usually necessary before the problem can be accurately identified and a treatment plan developed. Children's learning problems can be identified in a similar manner. In most public schools, specific individual ability and achievement tests to clarify learning problems are administered by school psychologists and educational diagnosticians.

What Do Evaluations Tell You?

As you continue on your advocacy journey, you must understand the exact nature of your child's disabling condition(s). How does the disability affect her? In what areas? How serious is it? What are her strengths and weaknesses? Does she need special education? What educational issues need to be addressed? How will you know if she is making progress? How much progress is sufficient? The answers to these questions will be found in the evaluations and tests that are administered to children and adolescents.

Many parents erroneously believe that they cannot understand the tests. They believe that this information is beyond their ability to understand or comprehend. Usually, their reasoning goes like this:

Gosh. I'm just a parent. I didn't even finish college. I don't have any training in education or special education so I can't understand that stuff!


The people who did that testing on my kid went to school for years to learn how to do that. Who am I to think I can understand it? I'm not a psychologist!

If you believe that you "can't" understand your child's testing, it's time to change your beliefs. You may be reading this article because your son or daughter is performing poorly in school --- or has been identified with learning problems --- and now believes that he or she "can't" read or write or do arithmetic. Your child must confront and overcome these erroneous beliefs about learning new or difficult material. And, so must you.

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Statistics: General Principles 

Statistics are simply ways to measure things and to describe relationships between things, using numbers. Part of the confusion that many people experience when they first begin to learn statistics is because of the unfamiliar terms and concepts. As we learned in our earlier discussion about measuring physical growth, there are several different ways to report the same information (inches, feet, yards, centimeters, etc.) In the beginning, this can be confusing.

First, let's look at another familiar example that many of us deal with regularly --- how to measure our car's gas mileage. Remember: When using statistics, we can use several different terms to describe the same concepts. If you want to describe your car's gas mileage, you can make any of the following statements:

  • My gas tank is half full.
  • My gas tank is half empty.
  • I am at the fifty percent mark.
  • My odometer shows that I have another 150 miles before the next fill-up.
  • My odometer shows that I have traveled 150 miles since I last filled the tank.
All of these statements accurately describe your car's consumption of gas.

With this information, you can make decisions. When will you need to buy more gas? You know that your car has a fifteen gallon gas tank. According to the gas gauge, your tank is slightly below the halfway mark. You've been driving in the city. You'll be driving on the highway for the rest of your trip. You have used a precise amount of gas and have a precise amount of gas left in your tank. You can describe and define this information in several ways --- gallons used, gallons remaining, miles driven, miles to go, percentage full, and so forth. Using the information above, you can do some simple math calculations and learn that your car averages between seventeen to twenty-three miles to a gallon of gas, depending on driving conditions.

Using this information or data, you can also measure change. If you compare your car's present or current mileage to the mileage you obtained last month, before you had your car tuned up, you can measure miles per gallon before and after the tune-up. In this way, you can measure the impact of the tune-up on your car's gas consumption. You can also compare your car's mileage performance to that of other vehicles.

Let's look at another common way in which we use tests and measurements. When you last visited your doctor, you mentioned that you were feeling tired and sluggish. Your doctor asked several questions, then recommended that you have some lab work. After reviewing the test results, the doctor explained that your blood glucose level was moderately elevated.

To lower your blood glucose level, the doctor recommended a plan of treatment that included a special diet and a daily program of moderate exercise. After a month, you return for a follow-up visit. More lab work is completed. If your glucose level has returned to normal, it is unlikely that you will require additional treatment. But, if your glucose level remains high, despite the diet and exercise program, you may need more intensive treatment. By measuring change after an intervention and using "appropriate objective criteria and evaluation procedures," you and your doctor can make rational decisions about your medical treatment.

Remember: The principles that enable you to compute your car's gas mileage and make medical decisions will also enable you to understand educational change. When you measure educational progress (just as when you measure your gas mileage and blood levels), the test scores can be reported and compared in several different ways.

Because educational test scores are often reported in different formats and compared in different ways, it is essential for parents and advocates to understand all of the scoring methods used in measuring and evaluating educational progress, including:

  • age equivalent scores (AE)
  • grade equivalent scores (GE)
  • standard scores (SS) and standard deviations (SD)
  • and percentile ranks (PR).
Knowledge about statistics will enable you to assess your child's progress or lack of progress in a particular educational program. Lack of progress is usually referred to as regression. Unfortunately, regression is a common educational problem that we will discuss in more detail later. You must learn how to recognize regression and reverse the downward spiral before your child is further damaged.

Statistics: Applied

Let's turn our attention to the performance of a group of children. You must understand how an individual child scores when compared with other children who are his age or in his grade --- and what this means.

First, we will examine a single component of physical fitness in a group of elementary school students. Our group or sample consists of 100 fifth grade students. These children are enrolled in a physical fitness class to prepare them to take the President's Physical Fitness Challenge. We will assume that the average chronological age (CA) of these children is exactly ten years, zero months. (CA=10-0) The children are tested in September, at the beginning of the school year.

To qualify as "physically fit," each child must meet several goals. Push-ups are one measure of upper body strength. Each child must complete as many push-ups as possible in a period of time. Each child's raw score is the number of push-ups completed. The term raw score is simply another way of describing the number of items correctly answered or performed.

After all of the fifth grade students complete the push-up test, their scores are listed. The results are as follows:

  • Half of the children completed ten push-ups or more.
  • Half of the children completed ten push-ups or less.
  • The average child completed 10 push-ups.
  • The average or mean number of push-ups completed by this class of 100 fifth grade students is 10.
  • Half of the children scored above the mean score of 10.
  • Half of the children scored below the mean or average score of 10.
  • 50 percent of the children scored 10 or above
  • 50 percent of the children scored 10 or below.
As we continue to analyze the children's scores, we see patterns:
  • One-third of the children scored between 7 and 10 push-ups.
  • One-third of the class completed between 10 and 13 push-ups.
  • Two-thirds of the children scored between 7 to 13 push-ups.
  • Half of the children (50 percent) completed between 8 and 12 push-ups.
  • The lowest scoring child completed 1 push-up.
  • The highest scoring child completed 19 push-ups.

Again, two-thirds of the children in this fifth grade class were able to complete between 7 and 13 push-ups. The remaining third of the children did fewer than 7 or more than 13 push-ups. Nearly all of the children --- 98 out of 100 --- were able to complete between 4 and 16 push-ups. Click here to view the bell curve chart.

The test results provide us with a sample of data. As we analyze the data in our sample, we can compare the performance of any individual child with that of the entire group. As we make these comparisons, the data will enable us to recognize any individual child's strengths and weaknesses when compared with the peer group of similar youngsters.

If we conduct an identical push-up test with children in other grades, we can compare our original group of 100 fifth grade children with other groups of youngsters --- children who are older, younger, in different grades, in different schools. If we gather enough information or data from other sources, we can compare our original group of fifth graders --- or an individual child within our group --- to a national population of children who are being tested for their upper body strength as measured by their ability to do push-ups.

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Measuring Progress: The Bell Curve 

In nature, traits and characteristics distribute themselves along theoretical curves. For our purposes, the most important curve is called the normal frequency distribution or bell curve. Because the percentages of areas along the bell curve are well-known and thoroughly researched, they become our frame of reference.

By using the bell curve, we can now develop an actual diagram or graph of the children's push-up scores. This map --- on the bell curve --- provides us with additional information. We can see what percentages of children were able to complete specific numbers of push-ups. When we use the bell curve, we can visually demonstrate where any particular child scores, when compared with other children who are the same age or in the same grade. Likewise, with educational test scores, we can visually demonstrate scores and change over time.

If we compare the push-up scores obtained by children who attend different schools, we can determine whether the physical fitness of children, as measured by their ability to do push-ups, varies in different schools, neighborhoods, states, or countries.

We can also measure progress over time --- with push-ups and with improvement in reading skills. Let's look at our class of fifth graders again. We want to gather information as to whether the physical fitness class is effective --- whether the children's fitness levels improve. How can we answer this question?

To measure the effectiveness of the fitness class, we will measure the children's number of push-ups before they take the class and compare this score with their score after they take the class. If the class is effective, we should see individual improvement and group improvement. Some children will have minimal improvement --- these children will fall further behind the peer group. Other children who performed below their peers may show significant improvement. Some children will improve so much that they now perform as well or better than the "average" youngster.

We will measure the children's progress on one or more occasions as they progress through the class. If the fitness class is "working," that is, if the children's' fitness levels are improving, then their ability to perform fitness skills should improve measurably over time. In our example, physical fitness improvement is being assessed using "appropriate objective criteria and evaluation procedures . . ." (34 C.F.R. §300.346)

Because of its enormous usefulness in measuring educational progress, we will return to the subject of the bell curve repeatedly throughout this article.

Understanding The Bell Curve

On all bell curves, the bottom or horizontal line is called the X axis. In our sample of fifth graders, the X axis represents "number of push-ups." And, on all bell curves, the up- and- down vertical line is called the Y axis. In our sample, the Y axis represents the number of children who earned a specific score (number of push-ups completed).

As you can see in the diagram (above), the highest point of the bell curve on the X axis equals a score of ten push-ups. You recall that more children completed ten push-ups than any other number. Thus, the highest point on this bell curve represents a score of ten. The next most frequently obtained scores were 9 and 11, followed by 8 and 12. This pattern continues out toward the extreme ends of the bell curve. In our example, the extremes occurred at 1 and 19 push-ups.

Using the bell curve, we can now chart each child's score and compare it to the score achieved by all 100 students in the class. Look at the bell curve above, and find 10 push-ups. We know that Amy completed 10 push-ups so her raw score was 10. Ten push-ups placed her squarely in the middle of the class. Half of the youngsters in Amy's class earned a score of 10 or more; half of the children scored 10 or less. If you look at the bell curve diagram (below), you see that Amy's score of 10 placed her at the 50% level. The individual's percent level is referred to as their percentile rank (PR). Amy's percentile rank is 50 (PR=50).

Erik completed thirteen push-ups. Looking at the bell curve above, you see that his score of 13 placed him at the 84th percent level. Erik's percentile rank is 84 (PR=84). Erik's ability to do push-ups placed him at the 84th position out of the 100 fifth grade children tested on our measure of upper body strength.

Sam completed seven push-ups. His raw score of 7 placed him at the (bottom) 16 percent. Sam's percentile rank was 16 (PR=16). Out of our sample of 100 fifth grade children, 84 children earned a higher score than Sam.

Larry completed 6 push-ups. We can convert his raw score of 6 to a percentile rank of 9 (PR=9). 91 children scored higher and 8 children scored lower than Larry in upper body strength as measured by the ability to do push-ups.

Oscar completed 2 push-ups. His raw score of 2 placed him in the bottom 1 percent of fifth graders tested (PR=1).

Nancy's raw score of 17 placed her at the upper 99 percent. We say that Nancy scored at the 99th percentile rank (PR=99).

You can see the relationship between the number of push-ups completed and the child's percentile rank (PR) reproduced in the table below: Click here to see the table

The bell curve is a powerful tool. When you use the bell curve, you can objectively compare any child's percentile rank to that of a group of children. You can also compare a single child's progress or regression when compared to the group.

Using the bell curve, you can compare a single child's score to the scores obtained by other children who are older or younger or in different grades.

Let's see how this works. Again, we will measure the children's upper body strength by the number of push-ups they can perform. In this case, we decide to evaluate all children in all the elementary grades, from Kindergarten through fifth grade. We will assume that the average chronological age of these elementary school children is exactly eight years (CA=8-0 years).

After we test the third graders, we find that the average or mean score of our sample of 100 eight year old third graders is 6 push-ups. This means that the "average" third grade child (who is 8 years old) can do 6 push-ups. We can also compare an individual child's score on arithmetic problems answered correctly with the average number answered correctly by children the same age.

How can we compare children from different groups? Let's look at Larry who was a member of our original group of fifth graders. Although the average fifth grader performed 10 push-ups, Larry only completed 6 push-ups. His raw score of 6 converts to a percentile rank of nine (PR=9).

When we compare Larry's performance to all elementary school students, we learn that Larry (a fifth grader) is functioning at the level of the average third grader --- who is also eight years old --- in the ability to do push-ups. Therefore, we see that Larry's age equivalent score is 8 years (AE=8-0) and his grade equivalent score is at the third grade level (GE=3-0).

 Fifth Grade Students: Push Up Scores
Child's Name Raw Score Percentile Rank
Oscar 3 1
Larry 6 9
Sam 7 16
Amy 10 50
Erik 13 84
Frank 15 95
Nancy 17 99

Look at the table above and find Amy. At the time of testing, Amy was 10-0 years old and in the fifth grade. She scored at the mean for her peers, i.e., 10 push-ups. Her grade equivalent score was fifth grade (GE=5-0) and her age equivalent score was 10.0 years (AE=10-0). If we tested a 20 year old person and found that this person was able to do 10 push-ups, then the 20 year old has an age equivalent score of 10-0 and a grade equivalent score of 5.0, i.e., the same score as Amy.

Look again at the table of scores above and find Frank's name. You see that Frank earned a raw score of 15 push-ups which converts to a percentile rank of 95 (PR=95). Frank's score looks great --- until we remember that Frank was "held back" three times. Although he is in the fifth grade, Frank is 13 years old!

With this new information, let's take another look at Franks' performance. The average score for 8th graders (who are 13 years old) is 15. Frank scored 15. Frank had a grade equivalent score of 8th grade (GE = 8.0) and an age equivalent score of 13 years (AE = 13-0). When we compare Frank with other children in his expected grade, we see that his achievement is in the average range. Frank is in the 95th percentile level when compared to fifth graders, not when compared to eighth graders.

Frank's case brings up some additional questions. Frank (age 13) was included in our sample of 5 th graders who had an average age of 10. When compared to this group of children who were younger than him, Frank scored at the 95% percentile rank (PR) level. Question: If we compare Frank's performance to that of children who are three years younger than him, will this comparison provide us with an accurate picture of his physical fitness? Answer: No.

In Frank's case, statistics inform us of two facts. First, we see that Frank performs at a superior level when compared with other children in his grade. Second, we see that he performs at an average level when compared with children who are his age.

When you evaluate the significance of data from tests, you must know how the scores are being reported. Test scores can be reported using percentile ranks, age equivalents, grade equivalents, raw scores, scale scores, subtest scores, or standard scores.

Remember: Although Frank's performance was superior for his grade, it was average for his age. If you did not know Frank's age and grade, you would have been misled as to Frank's actual achievement. But --- if Frank was an 8 year old 3rd grader, his scores would be in the superior range, using both age equivalent and grade equivalent measures.

The number of push-ups each child completed was his or her raw score. Let's assume that we want to obtain an overall fitness score. To obtain an overall or composite score, we will measure three skills (sit-ups, push-ups, a timed 50 yard dash) and obtain scores on each of these skills. In educational testing, the child's overall score (in reading, math, etc.) is often a composite of several subtest scores.

Next, we will develop a weighting system that will convert each child's raw score to a scale score. After we convert the raw scores to scale scores, we will be able to compare each of the three scores to each other (number of push-ups, number of sit-ups, seconds to complete the 50 yard dash). How do we convert raw scores into scale scores?

One way to convert scores is by developing a rank order system. In rank order scoring, the child who scores highest in an event (most push-ups, most sit-ups, fastest run) receives a scale score of 100; the lowest receives a score of 1. The other 98 children receive their respective "rank" as their scale score.

After each child's raw scores are converted to scale scores, we can easily compare an individual child to the group and to all children who are the same age or in the same grade. We can also compare an individual child's performance at different times, i.e. before and after completing the fitness course. Was the child able to do significantly more push-ups after taking the fitness course? Was the child reading better after receiving reading remediation?

Composite Scores

You can see that after we develop a global composite score, the individual child's raw scores on each of the three fitness subtests have less significance. This is exactly what happens with educational achievement and psychological tests. Most educational tests are composed of several subtests; the subtest scores are combined to develop composite scores. More about this shortly.

Let's look at how composite scores can be used and some of the problems that arise when we rely on them.

John is a member of our original group of 100 fifth graders. He has good muscular strength (he scored at the 70% PR level in push-ups and at the 78% PR in sit-ups). But, John is very slow and uncoordinated. In the 50 yard dash, he finished 2nd from the last out of the 100 children (PR=2).

How will John's composite fitness score be derived? In this example, we will average John's percentile rank scores on the three events. John's composite score is determined as follows: Add the percentile ranks of each event (70 + 78 + 2 = 150), then divide this score by the number of events (3). In John's case, 150 / 3 = 50. (Note: actually it is improper to average the percentile rank scores, you must use the standard scores or scale / subtest scores.)

John's composite score is 50. This composite percentile rank score of 50 places him squarely in the "average" range. Is John an "average" child? His individual scores demonstrated a significant amount of subtest scatter. When you analyze his three subtest scores, you see that he has specific strengths and a very severe deficiency. Despite his average composite score, John is not an average child! (Note: As noted above, the proper calculation is to use the standard scores. Thus the same analysis of John's composite score by using standard scores, is calculated to a standard score of 96.5 and percentile rank of 41 --- again, John appears to be an average child).

Let's look at another example of composite scores to see how they can mislead us. Oscar was at the 1 percent level in push-ups. But when the other fitness subtests were given, Oscar was the fastest child in the class scoring at the 99% level. He was average in sit-ups, scoring at the 50% level. Oscar's composite fitness score, using percentile ranking, is 50%. Is Oscar really an average child? Would he benefit from remediation to improve his upper body strength, as measured by push-ups? Oscar also a great deal of subtest scatter, i.e., from extremely weak upper body strength to superior speed.

Subtest Scatter

When subtest scores vary a great deal, this is called subtest scatter. If significant scatter exists, this suggests that the child has areas of strength and weakness that need to be explored.

How can you determine if significant subtest scatter is present? Most subtests have a mean score of 10. Most children will score + or - 3 points away from the mean of 10, i.e. most children will score between 7 and 13.

If the mean on a subtest is 10 (and most children score between 7 and 13), then scores between 9 and 11 will represent minimal subtest scatter. Lets assume that Child A is given a test that is composed of 10 subtests. The child's scores on the 10 subtests are as follows: on 4 subtests, the child scores 10, on 3 subtests, the child scores 9, and on 3 subtests, the child scores 11. In this case, the overall composite score is 10 and the scatter is very minimal. This child scored in the average range in all 10 subtests.

In our next example, we will assume that Child B earns 4 subtest scores of 10, 3 scores of 4, and 3 scores of 16. The child did extremely well on 3 tests, very poorly on 3 tests, and average on 4 subtests. Again, the child's composite score would be 10. Subtest scatter is the difference between the highest and lowest scores. In this case, subtest scatter would be 12 (16-4 = 12) Is this an "average" child? Because the child's scores demonstrate very significant subtest scatter, we need to know more about these weak and strong areas.

In educational situations, it is essential that parents understand the nature of the weak areas, what skills need to be learned to strengthen those areas, and how the strong areas can be used to help remediate the child's weak areas. The spread or variability between the subtest scores is called subtest scatter.

How do these concepts (composite scores and subtest scatter) relate to the information contained in your child's evaluations?

The results of educational tests given to children are often provided in composite scores. On the Wechsler Intelligence Scale for Children, Third Edition (WISC-III), three scores are usually provided --- a Verbal IQ (VIQ), a Performance IQ (PIQ), and a Full Scale IQ (FSIQ). Each of these IQs are composite scores. Both the Verbal and Performance IQ scores are composites of five different subtests, each of which measures a different area of ability. The Full Scale IQ is a composite of the Verbal and Performance scores --- which makes it a composite of ten different subtests. IQs between 90 and 110 are considered within the "average range."

If we rely on composite IQ scores, we may easily be misled -- with serious consequences. Katie is the 14 year old youngster whose situation was outlined earlier in this article. On the Wechsler Intelligence Scale for Children-III, Katie achieved a Full Scale IQ of 101. If the only number you had was her Full Scale IQ score, you would probably assume that her IQ of 101 placed her squarely in the "average range" of intellectual functioning. Is Katie an "average" child?

Remember: The Full Scale IQ score is actually a "composite" of the Verbal IQ and Performance IQ scores. Checking further, you learn that Katie's Verbal IQ is 114 and her Performance IQ is 86. IQ scores between 110 and 90 are considered "average." You see that there is a 28 point difference between Katie's Verbal and Performance IQ scores. If you did not have these additional two IQ scores, you might view Katie as an "average" child but you would be mistaken.

Katie's Verbal IQ of 114 translates into a percentile rank of 82 (PR=82). Her Performance IQ of 86 converts to a percentile rank of 18 (PR = 18). We see that Katie has a percentile rank fluctuation of 64 points (82-18=64) between her verbal and performance abilities. We will look at more of Katie's test scores shortly.

One of the commonly administered individual educational achievement tests is the Woodcock-Johnson Psycho-Educational Battery-Revised (WJ-R). The Woodcock-Johnson consists of a number of mandatory and optional subtests. The results obtained by the child on these different subtests are combined into composite or cluster scores. If we rely on composite or cluster scores, without examining the child's scores on the individual subtests, we can easily overlook obvious deficiencies and significant strengths. Relying on composite or 'cluster' scores can lead to faulty educational decision-making, having tragic consequences for children. To advocate effectively, parents must obtain all of the subtest scores on the tests that have been administered on their child.

When Apparent Progress Means Actual Regression

One serious concern that many parents have relates to the belief that their child is not making adequate progress in a special education program. How can parents determine if their perception is accurate? And, how can parents persuade school officials that the special education program being provided to the child needs to be strengthened?

Earlier in this article, we discussed how statistics can be used in medical treatment planning. We demonstrated how a medical problem was identified and the efficacy of treatment measured, using objective tests. In our example, the patient had pre- and post- testing as a means to determine whether or not the intervention was working. Based on the results of new testing, more medical decisions would be made --- to continue, terminate or change the treatment plan.

This practice of measuring change, called pre- and post- testing, has great relevance to educational planning. After the child's performance level is identified, we can re- test the child later to measure progress, regression, or whether the child is maintaining the same position within the group.

In this way, pre- and post- testing enables us to measure educational benefit (or lack of educational benefit). Using the scores obtained from pre- and post- testing, we can create graphs to visually demonstrate the child's progress or lack of progress in an academic area.

To see how this works, let's revisit our fifth grade fitness class. According to our earlier testing in September, Erik completed 13 push-ups which placed him in the top 84 percent of all youngsters in his class. After a year of fitness training, all of the fifth grade children were re-tested. When Erik was re-tested, he completed 14 push-ups.

    Question: Has Erik progressed?
    Answer: Yes and no.
The average performance of the fifth grade class improved by 2 push-ups (from an average raw score of 10 to an average raw score of 12). Erik's raw score increased by 1 push-up, from 13 to 14. So, we see that although Erik's age equivalent and grade equivalent scores increased slightly from the prior testing, his actual position in the group dropped from the 84 th to about the 75 th percentile level. While still ahead of his peers, Erik did regress.

What about Sam? Sam's push-up performance also improved, from a raw score of 7 to a raw score of 8. Although Sam's age equivalent and grade equivalent scores increased slightly, he also regressed. According to the new scores, his percentile rank dropped from the 16 percentile to about the 9 th percentile rank. Sam is continuing to fall further behind his peer group.

Let's assume that we test Sam again when he re-enters school in the fall. Now, we have three sets of test data (beginning 5 th grade, end 5 th grade, beginning 6 th grade). Has Sam's score changed? If his percentile rank continues to drop, Sam is experiencing regression. We need to know how long will it take for Sam to recoup the skills he lost during the summer. Regression and recoupment are primary issues in determining the child's legal need for extended school year services (ESY) during the summer.

Norm Referenced versus Criterion Referenced Tests

Most standardized tests are either norm referenced or criterion referenced.

When we evaluated our sample group of fifth graders, we compared each child's performance to the norm group of fifth graders. Both Erik (raw score of 13, percentile rank of 84) and Sam (raw score of 7, percentile rank of 16) were referenced or compared to this norm group of fifth graders. To evaluate benefit, we looked at the norm group and the individual child's relative position in that group at the time of the first and second tests. We computed each child's change in position, i.e. progress or regression.

In our example, we also referenced the criteria of number of push-ups completed. A criterion reference analysis determines whether or not a child meets certain criteria (without reference to a norm group.) For example, at the beginning of the year, Sam completed 7 push-ups. If the criteria for success was 8 push-ups, then Sam failed to reach that goal. Let's assume that Sam received a year of physical fitness remediation; after that year, Sam completed the 8 push-ups. Does Sam now met the criteria for success? The answer to this question depends on whether the criteria have increased now that Sam is a year older.

Another factor complicates this picture. We know that Sam's' peer group completed 10 push-ups at the beginning of the year and 12 at the end of the year. Definitions of success are affected by the passage of time. If we rely on criterion referenced measures, we can be misled as to whether the child is falling further behind the peer group. We need to know exactly what the criterion is and what this means when the child is compared to a norm group.

Standard Deviation

Percentile ranks are computed by determining the mean score and the amount of variation of all scores around the mean score. Are the scores bunched around the number 10 in a tight uniform distribution? Are the scores evenly distributed? Do they peak and taper slowly in our earlier bell curves, or do they bunch at the ends, without any scores in the middle? In other words, is there a great variance, with the scores spread over a wide range with two or more peaks, or is there a normal bell curve distribution of scores?

On our push-up test, most of the 5th grade children earned scores around 10 push-ups, with an even distribution above and below 10 push-ups. But, if one-half of the children completed 5 push-ups, one-fourth completed exactly 14 push-ups, and the remaining one-fourth completed 16 push-ups, then the average or mean number of push-ups would still be 10. One-half of the children would have scored above 10 and one-half below 10.

In this case, the distribution is not evenly distributed in a smooth curve above and below the score of 10. In fact, the variance is very large and would present a highly unusual curve with a peak at 5, a drop to zero between 6 and 13, then a jump at 14, a drop at 15, another jump at 16. This distribution of scores would not present a normal bell curve distribution. Educational and psychological tests are designed to present normal bell curve distributions with predictable patterns of scores.

We simply need to know the mean and standard deviation of the test. In most educational and psychological tests, the mean is 100 and the standard deviation is 15. (Mean = 100, SD = 15) In most subtests, the mean is 10 and the standard deviation is 3. (Mean = 10, SD = 3) Average scores do not deviate far from the mean. As scores fall significantly above or below the mean, they are referred to as being a certain value or distance from the mean, e.g., 1 or 2 standard deviations from the mean.

In all tests, the mean is at 0 (zero) standard deviations from the mean. The next marker on the bell curve is +1 and -1 standard deviations from the mean, followed by 2 standard deviations from the mean. To interpret your child's test scores, you will need to know the test instrument's mean score and standard deviation score.

Using our original push-up example, the mean score was 10 push-ups and the standard deviation (SD) was 3 push-ups. This push-up example is identical to the subtest scores in almost all standardized educational and psychological testing.

REMEMBER: With most subtest scores, the mean is 10, and the standard deviation is 3.

One standard deviation above the mean is 10 plus 3, i.e. 10 + 3 = 13. One standard deviation below the mean is 10 minus 3; i.e. 10 - 3 = 7. One standard deviation above the mean always falls at the 84 percent level (PR = 84); one standard deviation below the mean is always at the 16 percent level (PR = 16). Two SD's above the mean is always at the 98 percent level (PR = 98); and two SD's below the mean are always at the 2 percent level (PR = 2).

Looking at actual test scores, we may see that the child scored "one standard deviation below the mean" on a particular test or subtest If the score is one standard deviation below the mean, then the child's percentile rank is 16.

REMEMBER: The subtest scores of most tests used with our children have a mean of 10 and standard deviation of 3. If a child scores 7 on a subtest, this means that the child scored at the 16 th percentile. A subtest score of 13 means that the child scored at the 84 th percentile.

Standard Scores

One of the most difficult concepts for most parents to grasp is that of standard scores. Since many educational test scores are given in standard scores, it is essential for parents to understand what they mean.

At an IEP meeting, a parent may be told that the child earned a standard score of 85 in one area, a standard score of 70 in another area. Most parents are relieved when they get this news --- because they believe that these numbers are similar to grades with 100 as the top score and 0 as the lowest. This is absolutely incorrect! Standard scores are NOT like grades.

In standard scores, the average score or mean is 100, with a standard deviation of 15. The average child will earn a standard score of 100. If a child scores 1 standard deviation above the mean, the standard score is 100 plus 15; i.e. 100 + 15 = 115. If the child scores 1 standard deviation below the mean, this is 100 minus 15, i.e. 100 - 15 = 85.

Since a standard score of 115 is 1 standard deviation above the mean, it is always at the 84 percent level. Since a standard score of 85 is 1 standard deviation below the mean, it is always at the 16 percent level. A standard score of 130 (+2 SD) is always at the 98 percent level. A standard score of 70 (-2 SD) is always at the 2 percent level.

Remember Katie? Earlier, we learned that on the Wechsler Intelligence Scale, Katie earned a Full Scale IQ of 101. Later, we saw that this score was misleading because Katie's Verbal IQ score was 114 while her Performance IQ score was 86. The psychologist found that Katie scored 2 standard deviations above the mean on the Similarities subtest of the Wechsler Intelligence Scale for Children, 3rd Revision (WISC-III). What does this mean?

You are learning that a score of 2 standard deviations above the mean places the child at the 98th percent level on the area being measured. Since the Similarities subtest of the WISC-III measures intellectual reasoning power, Katie's intellectual reasoning power is at the 98 percent level.

The psychologist also found that Katie had a standard score of 68 --- which was 2.5 standard deviations below the mean -- on the spontaneous writing sample of the Test of Written Language (TOWL-III). Two SD's below the mean is at the two percent level. With your new knowledge, you know that Katie's ability to produce spontaneous writing samples was actually lower than the one percent level.

When we first introduced Katie, we posed two questions:

1. Do these two test scores help to explain the academic problems Katie is having?

2. Do her test scores tell us anything about her moodiness and her intense dislike of school?

Katie's intellectual reasoning ability places her at the top 98 percent of all youngsters her age. However, her ability to convey her thoughts in writing is below the one percent level. If Katie is very bright but is unable to convey her knowledge to her teachers on written assignments and tests, would you expect her to feel frustrated and stupid? Do you question why, after years of frustration, Katie is angry, depressed and now wants to quit school?

Wrightslaw Rules 

All educational and psychological tests that report scores using percentile ranks or standard scores are based on the bell curve. To interpret the tests results, you should know the mean and the standard deviation. The Wechsler, Woodcock-Johnson, Kaufmann, and most other standardized tests use this format.

  • Since most educational and psychological tests use standard scores (SS) with a mean of 100 and a standard deviation of 15, a standard score of 100 is at the 50% percentile rank (PR) level. A standard scores of 85 is at the 16 % PR level. A standard score of 115 is at the 84% PR level.
  • Most educational and psychological tests use subtest scores with a mean of 10 and standard deviation of 3. A subtest score of 10 is at the 50% PR level. Subtest scores of 7 and 13 are at the 16% and 84% PR levels.
  • One half of all children fall above and one half of all children fall below the mean of 50% which is also represented as a standard score of 100. A standard score of 100 = PR 50.
  • Two-thirds of all children are between + 1 and - 1 standard deviations from the mean.
  • Two-thirds of all children are between the 16% and 84% percentile ranks. (84 minus 16 = 68)
  • A standard deviation of -1 is at the 16% level. Zero is at the 50% level. +1 SD is at the 84% level.
  • A standard score of 85 is at the 16% level; a SS of 100 is at the 50% level; a SS of 115 is at the 84% level.
  • A standard deviation of -2 is at the 2% level. A SD of +2 is at the 98% level.
  • A standard score of 70 is at the 2% level. A standard score of 130 is at the 98% level.
  • A standard score of 90 is at the 25% level. A standard score of 110 is at the 75% level.
  • One half of all children fall between the 75% level and 25% level. (75-25 = 50)
  • One half of all children achieve standard scores between 90 to 110.
  • A percentile rank score between 25% and 75% is the same as a standard score of between 90 to 110 --- and are usually considered to be within the "average range."

Understanding Test Data 

The results of most educational tests are reported using standard scores. Parents must know how to convert standard scores into percentile ranks. Using the table below and bell curve above, you can convert any standard score into a percentile rank score. The earlier push-up example used standard educational scores. Click here to view the table

Other Tests: Means and Standard Deviations

Adding to the confusion about tests is the fact that test scores are sometimes reported differently. For example, test scores may be reported as "Z Scores." Z scores are simply standard deviation scores of one with a mean of zero (Mean = 0, SD = 1, instead of a mean of 100 and SD of 15 as we found with standard scores).

If you know that a particular child earned a Z score of -1, then you also know that the child's score was one standard deviation below the mean, which is a percentile rank of 16. If you convert this score, using the standard score format with a mean of 100 and a standard deviation of 15, you will see that a z score of -1 is the same as a standard score of 85.

Another test format uses T Scores. With T scores, the mean is 50 and each unit of standard deviation is equal to 10. A T score of 60 is the same as a Z score of +1. A T score of 60 and a Z score of +1 are equal to a percentile rank of 84. A T score of 70 is equal to a Z score of +2, a standard score of 130, and a percentile rank of 98.

Another measure is a Stanine test. In Stanine tests, the mean is five and the standard deviation is 2.

Specific Tests

Since tests are always in a state of change with new versions being produced, we will not attempt to review and describe each test. There are a number of parent-oriented publications that you can refer to. Interested people may ask the examiner to photocopy relevant portions of the manual for you. Examiners cannot copy actual test questions for you, but may be able to copy the instructions and explanations. This is your best source of current test information.

Earlier in this article, you learned that both the Verbal and Performance IQ scores are actually composites or averages of five different subtests. Each of the separate subtests measures very different abilities. Let's analyze Katie's subtest scores to see what else we can learn from them.

Wechsler Intelligence Scale for Children, Third Edition (WISC-III) 
 Verbal Subtests   Performance Subtests
 Information  10  Picture Completion  6
 Similarities  16  Coding  4
 Arithmetic  11  Picture Arrangement  10
 Vocabulary  13  Block Design  12
 Comprehension  12  Object Assembly   7
 (Digit Span)   8  (Symbol Search)  6
 Verbal IQ = 114
 Performance IQ = 86

Subtests of the Wechsler Intelligence range from a low score of 1 to a maximum score of 19. As you learned earlier, these subtests have a mean of 10 and a standard deviation of 3. A subtest score of 7 is one standard deviation below the mean (-1 SD) which is the same as a percentile rank of 16 (PR = 16). You can also convert the subtest score of 7 into a standard score of 85 which has a percentile rank of 16.

When we discussed subtest scatter, we saw that variation among subtest scores is a valuable source of information. Look at Katie's subtest scores. She has significant scatter, from a high score of 16 on Similarities (98 percentile) to a low score of 4 (2 percentile) on Coding.

As a parent, you need to understand what the various subtests measure. When we discussed Katie's test scores, you learned that Similarities subtest is highly correlated with abstract reasoning. The Coding subtest measures visual- perceptual mechanics. The Coding subtest is highly correlated with reading achievement but has little relation to abstract reasoning.

Question: Which Wechsler subtest is most closely correlated to intellectual horsepower and reasoning ability?

Answer: The Similarities subtest.

Question: Which subtest measures a child's ability to decode visual symbols?

Answer: The Coding subtest measures decoding of visual symbols.

The Psychological Assessment Resources, Inc. describes each WISC-III subtest as follows:

Information: factual knowledge, long-term memory, recall.

Similarities: abstract reasoning, verbal categories and concepts.

Arithmetic: attention and concentration, numerical reasoning.

Vocabulary: language development, word knowledge, verbal fluency.

Comprehension: social and practical judgment, common sense.

Digit Span: short-term auditory memory, concentration.

Picture Completion: alertness to detail, visual discrimination.

Coding: visual-motor coordination, speed, concentration.

Picture Arrangement: planning, logical thinking, social knowledge.

Block Design: spatial analysis, abstract visual problem-solving.

Object Assembly: visual analysis and construction of objects.

Symbol Search: visual-motor quickness, concentration, persistence.

Mazes: fine motor coordination, planning, following directions.

intelligence testing usually includes a measure of a visual motor speed (as in the Coding subtest) and a measure of intellectual reasoning ability (as in the Similarities subtest). To develop an accurate picture of your child's strengths and weaknesses, you need to understand what the various subtests actually measure.

When subtest scores are in parentheses, this means that these scores are not computed as a part of the overall composite score. If you look at Katie's scores, you will see that (Digit Span) and (Symbol Search) are in parentheses. On the WISC-III, the Digit Span, Symbol Search and Mazes subtest scores are not included in the Verbal, Performance and Full Scale IQ scores. They are used to develop other composite scores.

More than half of all children with disabilities served under the special education law have learning disabilities and/or an attention deficit disorder. The most commonly administered tests fall under three categories: intellectual; educational; and projective personality tests.

In most cases, the intelligence test given is the WISC-III and/or the Stanford-Binet. Specific training and education is required before a test publisher will allow a diagnostician to administer the WISC-III. The Woodcock Test of Cognitive Abilities measures specific cognitive areas. This test may be administered by an educational diagnostician and does not require the same high level of training and certification to administer.

Other Tests

The National Information Center for Children and Youth with Disabilities (NICHCY) has published a comprehensive free article entitled "Assessing Children for the Presence of a Disability" by Betsy B. Waterman, Ph.D. It is recommended that parents read this article to further their understanding of the assessment process.

In an issue of The International (Orton) Dyslexia Society's newsletter Perspectives, Dr. Jane Fell Greene was asked about the proper tests to use with dyslexic and learning disabled children.

Dyslexia is difficulty with language. Dyslexics experience problems in psycholinguistic processing. They have difficulty translating language to thought (reading or listening), or thought to language (writing or speaking). Although psychological, behavioral, emotional or social problems may result from dyslexia, they do not cause dyslexia. One test is inadequate: a battery is required. Typical psychoeducational tests were not designed to identify dyslexia.
Dr. Greene recommended using the Detroit Tests of Learning Aptitude as a global test that primarily tests verbal and non verbal language. "It measures the level at which the individual would perform if appropriate interventions were implemented (as is required by federal law)."

The article recommended additional tests by age group. The tests for preschool and kindergarten were the Test of Phonological Awareness, Tests of Early Written Language, Test of Early Reading Ability, and the Preschool Evaluation Scale. For primary years, the following were recommended - Test of Phonological Awareness, Test of Language Development, Peabody Individual Achievement Tests, Gray Oral Reading Test, PIAT Test of Written Expression, and the Wide Range Achievement Test. For elementary students Dr. Greene recommended the Test of Language Development, the Peabody Individual Achievement Test, Gray Oral Reading Test, PIAT Test of Written Expression and the Wide Range Achievement Test. For the adolescent and adult she recommended the Test of Adolescent and Adult Language, the Peabody Individual Achievement Test, the Gray Oral Reading Test, the PIAT Test of Written Expression and the Wide Range Achievement Test. The Detroit was recommended for all age levels.

Another area of assessment involves projective personality testing. Projective personality tests help to assess the child's mental state, degree of anxiety, and areas of stress. They can be useful in showing that a child who is viewed as emotionally disturbed is actually a normal child who is intensely frustrated about educational problems. Children experience great frustration and unhappiness when they cannot succeed in school. If placed in a healthier environment where they are able to learn, many "emotional problems" disappear.

There are many other types of tests and "surveys." Children who have difficulty processing information and whose tests show great scatter may benefit from a neuropsychological evaluation. Neuropsychological evaluations include tests that assess specific neurological issues that affect learning. Other measures include surveys and questionnaires that provide norm reference data, most often about behavior, how children see themselves, and how parents and teachers view them.

REMEMBER: To fully understand your child's test scores, you must know the mean, the standard deviation, and the child's specific score on the test, reported as either a standard score or a percentile rank. After you have the standard score or percentile rank, you can derive the other score.
Many test publishers also provide age equivalent and grade equivalent scores for specific raw scores.

After you master the information contained in this article, you will be able to convert test scores into easily understood numbers. You will be able to measure your child's educational progress. After you master this material, the feelings of helplessness and confusion that you have experienced at earlier school meetings will dissipate. You will become an authority in discussing your child's test score history and the significance of the data.

Private Sector Evaluations

In most of our cases, we do not rely on public school testing. Instead, we secure testing from private sector diagnosticians, child psychologists, school psychologists, and educational diagnosticians who are familiar with and able to administer a number of the multitude of tests that are available. We find that public school staff are often limited in the types of tests available for them to use and are unable to probe adequately, despite unusual scatter in a subtest profile.

Many private diagnosticians are eager to help parents learn how to chart out the child's test history. Assume that your child was tested three years ago on the WJ-R Test and scored at the 10% level in word identification, at the 60% level in passage comprehension and had a global composite reading score of 35%. After three year of special education where the child was presumably receiving remediation in reading, the child is retested privately. Subsequent testing by the expert discloses that your child is now at the 5% level in word identification and at the 45% level in passage comprehension, with a composite reading score of 25%. Technically, the earlier composite scores of 35% and 25% fall within the "average range." If you prepare a chart that demonstrates this regression, you may be able to convince school personnel to add true reading remediation to your child's educational program.

Individualized Education Programs

You should also obtain our book Wrightslaw: Special Education Law. The book (available from the Wrightslaw store and by fax and mail)  contains the complete federal statute (IDEA-97), the federal special education regulations, and Appendix A, the appendix that explains IEPs. 

You should also obtain the special education regulations from your State Department of Education. The language in the State's publication should be similar to the Federal Regulations.

By using this article and our law book, you will be able to write IEP's that contain measurable objectives.

For example, in an IEP that includes keyboarding, a typical public school IEP will measure typing success by using "teacher observation" at an 80 percent success rate. Your IEP will state that by December, 1996, on a five minute timed typing test of text, your child will be able to type at fifteen words per minute with one minute deducted for each error. By June, 1997, on a five minute timed typing test of text, your child will be able to type at thirty words per minute with five words per minute deducted for each error. This objective includes "Appropriate objective criteria and evaluation procedures and schedules, for determining, on at least an annual basis, whether the short term instructional objectives are being achieved." 34 C.F.R. Section 300.346

Parent's To Do List 

1. After you complete this article, make a list of all the times when your child has been tested. Arrange your list in chronological order. Include the names, dates, and scores of each test that has been administered to your child more than once.

2. Begin your list with the test or tests that have been administered most frequently. In many cases, that will be the Wechsler Intelligence Test and the Woodcock-Johnson and/or Kaufmann Educational Achievement Tests.

3. Write down all of the scores from the first administration of a test battery. Convert these scores to percentile ranks. Complete the same process with the most recent testing of the same battery. Compare the results. You should be able to determine whether your child is being remediated (catching up), staying in the same position, or falling further behind the peer group.

4. Dig for the standard scores or percentile rank scores in your child's file. You may find that some scores are only reported in "ranges" (i.e., high- average, low-average) or in grade equivalent or age equivalent scores. If the standard scores are not available, you should ask for them. When you request the data in standard score format, the school staff may be surprised but they should be able to comply with your request.

5. Take the most glaring deficiencies where your child has shown minimal progress or even regression and chart out the test results. If you do not have a computer, use graph paper. Software programs like Excel and PowerPoint allow for dramatic visual presentations of test data. If this is too difficult or confusing, consult with an expert. Gather your material --- your bell curve chart and standard score / percentile rank chart, your list of test scores, and your child's evaluations, and consult with a private sector psychologist or educational diagnostician who can explain the significance of the scores using percentile ranks.

6. Ask the professional to use the bell curve chart that includes standard scores, standard deviations and percentile ranks. Be sure that you have a photocopy of the bell curve so you can take it home to study. If the professional is willing, it may be helpful to tape record this portion of the session so that you can go back over it at home with the test scores in front of you.

7. Contact your state's Department of Education and request all publications about special education and IEPs, along with your state regulations.

8. Download our companion article, "Your Child's IEP: Practical and Legal Guidance for Parents and Advocates." 

For the professional, attorney, and the curious parent, an excellent book about tests and their meaning is Assessment of Children (currently being revised) written and published by Jerome M. Sattler, Publisher, Inc., P. O. Box 151677, San Diego, CA 92175. You can order this book from Dr. Sattler (619-460-3667) or from The Psychological Corporation (800-228-0752), or from the Advocate's Bookstore at Wrightslaw. On page 17 of Dr. Sattler's book, you will find a Bell Curve with percentile ranks for the Wechsler IQ tests, subtest scores, and most other tests that are used with special education children.

Go to:
https://www.wrightslaw.com/bellcurvepicture.pdf and https://www.wrightslaw.com/bellcurveandstandardscore.pdf where you can download and print bell curve charts and a list of standard scores, scale / subtest scores, standard deviation and percentile ranks! 

Make several prints of both. You'll be surprised at how often you'll  refer to them. Make copies for your friends.

Learn More About Tests and Assessments, See our Slide Show - Educational Progress Graphs 

Don't forget to download Your Child's IEP: Practical and Legal Guidance for Parents and Advocates.

Wrightslaw: From Emotions to Advocacy

Chapter Ten and Eleven of our newest book, Wrightslaw: From Emotions to Advocacy, focus exclusively on the information in this file. Bell curve charts, graphs, and many other visual aides to understanding Tests and Measurements are contained in those two chapters.

Chapter Eleven is titled: SMART IEPs and discusses the drafting of IEPs that are Specific, Measurable, use Action words such as to be Able to . . . , are Realistic, and Time Specific.

For more information about our new book, Wrightlaw: From Emotions to Advocacy, and the website, (www.fetaweb.com) that tracks the book, please go to:

URL: https://www.wrightslaw.com/bks/feta/feta.htm

Good Luck!

About the Author

Pete Wright represented Shannon Carter before the United States Supreme Court in Florence County School District Four v. Shannon Carter, 510 U.S. 7, (1993). In Carter, the Court issued a unanimous decision on November 9, 1993, just thirty-four days after oral argument. 

See CNN artist Peggy Gage's sketch of oral argument in the Carter case. Read the transcript of oral argument in Florence County School District Four v. Shannon Carter. Read the decisions in the Carter case. 

Revised:  November 26, 2001

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